To find the equation of the hyperbola, we proceed as follows:

Step 1: Foci of the ellipse

The given ellipse equation is:

$\frac{x^2}{100} + \frac{y^2}{36} = 1$

This is an ellipse centered at the origin $(0, 0)$ with the following properties:

• Semi-major axis: $a = \sqrt{100} = 10$
• Semi-minor axis: $b = \sqrt{36} = 6$

The distance from the center to each focus is given by:

$c = \sqrt{a^2 - b^2} = \sqrt{10^2 - 6^2} = \sqrt{100 - 36} = \sqrt{64} = 8$

Thus, the foci of the ellipse are located at $(\pm 8, 0)$.

Step 2: Hyperbola shares the same foci

For a hyperbola, the relationship between $a$, $b$, and $c$ is given by:

$c^2 = a^2 + b^2$

The eccentricity of the hyperbola is given as $e = 4$. The formula for eccentricity is:

$e = \frac{c}{a}$

Rearranging to find $a$, we get:

$a = \frac{c}{e}$

Substituting $c = 8$ and $e = 4$:

$a = \frac{8}{4} = 2$

Step 3: Solve for $b^2$

Using the hyperbola relationship $c^2 = a^2 + b^2$, substitute $c = 8$ and $a = 2$:

$8^2 = 2^2 + b^2$

$64 = 4 + b^2$

$b^2 = 64 - 4 = 60$

Step 4: Write the hyperbola equation

Since the foci lie on the $x$-axis, the hyperbola opens horizontally. The standard equation of a horizontally opening hyperbola is:

$\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Substituting $a^2 = 4$ and $b^2 = 60$:

$\frac{x^2}{4} - \frac{y^2}{60} = 1$

Final Answer:

The equation of the hyperbola is:

$\frac{x^2}{4} - \frac{y^2}{60} = 1$

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Would you like a detailed derivation or additional explanations?
Here are 5 related questions to explore further:

1. What happens to the shape of a hyperbola if its eccentricity increases further?
2. How is the equation of a hyperbola derived from its geometric definition?
3. Can you find the equation of the conjugate hyperbola of the given hyperbola?
4. What are the asymptotes of the hyperbola $\frac{x^2}{4} - \frac{y^2}{60} = 1$?
5. How do the properties of ellipses and hyperbolas differ geometrically and algebraically?

Tip: Always verify eccentricity and the foci of conic sections to ensure correctness when switching between ellipses and hyperbolas.